I personally would lean towards a 223, especially if you are concidering it to be a SHTF weapon as a secondary use. 22-250 ammo is common, 204 is less and 223 can be found everywhere that sells ammo. The 223 will do everything the others will do if you do your part.
As tango mentioned, 204 is a bit light for human targets and IMO 223 is light but that's another discussion.

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If the pain is lacking so is the discipline...

"the only 911 call I need is chambering a round" - Mr. Muller, MO car dealer

lots of pros and cons, to me after all the pros and cons, the .22-250 does the best, .223 cheaper and more availble, .204 a small bullet at long range. choose your poison but for me the .22-250 will kill hogs, deer, and any varmint in this part of Texas and is the first rifle I pull out of my safe any time of the year. I own lots of rifles in lots of calibers but no .204 yet, my .17 hmr covers that area for now in short range and the .22-250 covers the rest. after that my 6.5 x55 or .308 or 7-08 are all I shoot anymore, all are easy on the shoulder.